Emergent hydrodynamics and anomalous transport in quantum lattices

4 problems

• Controlled efficient computation of high-temperature diffusion constants

  Open

  In a generic chain of interacting quantum spins at high temperature, energy and spin spread diffusively at a definite rate, but no method computes this rate with a guaranteed error at reasonable cost. Exact simulation becomes exponentially expensive because the quantum state grows highly entangled.

• Asymptotic transport exponent in a strongly tilted two-dimensional Hubbard lattice

  Partially resolved

  When a lattice of atoms is strongly tilted, the centre of mass of the atoms can barely move, so density spreads more slowly than ordinary diffusion. Experiments see this slowing, but whether it is the final law at the longest times and distances is not established.

• Full counting statistics of spin superdiffusion in the Heisenberg chain

  Partially resolved

  In the spin-$1/2$ Heisenberg chain at high temperature, magnetization spreads faster than ordinary diffusion, with distance growing as time to the power $2/3$. The spreading has the scaling of a classical surface-growth model called KPZ (Kardar-Parisi-Zhang), but the full statistics does not match it, and the correct description is unknown.

• Does Heisenberg-chain superdiffusion survive symmetric integrability-breaking perturbations?

  Contested

  In the solvable Heisenberg spin chain, magnetization spreads faster than ordinary diffusion. If the chain is perturbed so that it is no longer solvable but keeps its rotation symmetry, it is unclear whether the fast spreading persists forever or eventually becomes ordinary diffusion.